Hoeffding's Lemma proof uses Taylor expansion with this statement:

From Taylor's theorem, for some $ 0\leq \theta \leq 1$

$ L(h) = L(0) + h L'(0) + \frac{1}{2} h^2 L''(h\theta) \leq \frac{1}{8}h^2 $

Why does it use $0$ in the first two terms and $h\theta$ in the last? But as I know they must be same in the Taylor.


1 Answer 1


This is the mean-value form of Taylor's theorem:

$$f(x)=f(0)+xf'(0)+\frac{x^2}{2}f''(c)$$ where $c$ is between $0$ and $x$

Take $x=h$ and $c=h\theta$

  • $\begingroup$ thanks. Is it related to Lagrange error? and can you explain a good document for mean-value form of Taylor's theorem? $\endgroup$
    – Tavakoli
    Commented May 27, 2022 at 9:03
  • $\begingroup$ Any decent Calculus textbook. $\endgroup$
    – whuber
    Commented May 27, 2022 at 13:34
  • $\begingroup$ The derivation is further down on the Wikipedia page I linked: en.wikipedia.org/wiki/… $\endgroup$ Commented Jun 1, 2022 at 21:25

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